Number & the laws of arithmetic
Order of operations
6 + 7 × 9 has one right answer, and it is not the one you get by working left to right. Multiplication goes first.
The order
When a calculation mixes operations, follow this order: 1. BRACKETS 2. MULTIPLICATION and DIVISION (left to right) 3. ADDITION and SUBTRACTION (left to right) Multiplying and dividing always come before adding and subtracting.
Why order matters
6 + 7 × 9 Left to right gives 13 × 9 = 117. That is WRONG. The correct way does 7 × 9 = 63 first, then 6 + 63 = 69. Without an agreed order, the same calculation would mean different things to different people. The rule is what makes everyone get 69.
Brackets come first
Brackets override everything. (6 + 7) × 9 = 13 × 9 = 117 6 + 7 × 9 = 6 + 63 = 69 Same numbers, different answers — and the brackets are the only difference. Use them whenever you want a different order.
Working left to right within a level
When operations are at the same level, work left to right. 54 ÷ 9 ÷ 3 = 6 ÷ 3 = 2 (not 54 ÷ 3 = 18) 20 − 5 + 3 = 15 + 3 = 18 (not 20 − 8 = 12) This catches people out with subtraction especially — do not jump ahead to the addition.
Worked examples
6 + 7 × 9
- Multiplication comes before addition.
- 7 × 9 = 63.
- Then 6 + 63.
Answer: 6 + 7 × 9 = 69
14 − 2 × 7
- Do the multiplication first: 2 × 7 = 14.
- Then 14 − 14.
Answer: 14 − 2 × 7 = 0
(6 + 7) × 9
- Brackets first: 6 + 7 = 13.
- Then 13 × 9.
Answer: (6 + 7) × 9 = 117
54 ÷ 9 ÷ 3
- Both are divisions, so work left to right.
- 54 ÷ 9 = 6.
- Then 6 ÷ 3.
Answer: 54 ÷ 9 ÷ 3 = 2
Filling in an operation
- Make 5 ▢ 9 ▢ 3 = 8.
- Try division first: 9 ÷ 3 = 3.
- Then 5 + 3 = 8.
Answer: 5 + 9 ÷ 3 = 8
Where you see this in real life
- Calculators follow this order, which is why typing a sum in wrongly gives an unexpected answer.
- Working out a bill of 3 items at $7 plus a $6 delivery charge: 6 + 3 × 7.
- Spreadsheet formulas use exactly the same rules.
Key words
- Operation
- Something you do to numbers, such as + or ×.
- Brackets
- Symbols showing what to work out first.
- Order of operations
- The agreed sequence for working out a calculation.
- Product
- The result of multiplying.
- Sum
- The result of adding.
- Left to right
- The direction to work when operations are equal.